91Ó°ÊÓ

In environmental science, understanding how fish populations sustain themselves involves calculating reproduction functions. The reproduction function represents how a population grows. In our example with rainbow trout, the function is given as:

• \[ f(p) = 50 \sqrt{p} \]

This function indicates the rate at which the trout population reproduces. A key factor here is understanding what the variables mean:

• \( p \) is the population size in thousands.
• \( f(p) \) represents the reproduction rate.

This function tells us that at any population \( p \), the reproduction rate grows with the square root of \( p \). Therefore, larger populations reproduce more until a certain point. Recognizing this value helps in determining the maximum sustainable yield.

Critical points are values in a function where the derivative equals zero or does not exist. These points can indicate maximum or minimum values of a function, which are crucial in determining optimal population sizes.
In our exercise, we need to find the critical point of the yield function, which combines the reproduction rate and the available population:

• \[ Y(p) = f(p) - p = 50 \sqrt{p} - p \]

To find the critical points, we set the derivative \( Y'(p) \) to zero:

• \( Y'(p) = \frac{25}{\sqrt{p}} - 1 = 0 \)

This calculation shows us where the surplus yield is maximized, important for sustainable fishing practices.

Derivative analysis helps find where a function increases or decreases. It involves calculating derivatives, which are formulas expressing rates of change.
To identify maximum or minimum points, we analyze first and second derivatives:

• First Derivative: \( Y'(p) = \frac{25}{\sqrt{p}} - 1 \)
• Second Derivative: \( Y''(p) = -\frac{25}{2p^{3/2}} \)

For our function \( Y(p) \), setting the first derivative equal to zero allows us to find the population values around the critical points.
The negative second derivative tells us the function \( Y(p) \) is concave downward at the point \( p = 625 \), confirming this is a maximum point. Thus, \( p = 625 \) is where sustainable yield is maximally efficient.

The concept of sustainable surplus is pivotal in resource management. It represents the difference between what is produced (reproduced in this case) and what is needed to maintain the population.
In the rainbow trout example, sustainable surplus is captured by the function \( Y(p) = f(p) - p \). Here, \( f(p) \) is the reproduction function, while \( p \) is the population size.

• When \( p = 625 \), the reproduction function equals \( 1250 \), and the sustainable yield is:
• \[ 1250 - 625 = 625 \]

This result tells us that at the maximum sustainable yield, the trout population can provide an additional surplus of 625 (in thousands), ensuring fishing activities can persist without depleting the population.
Understanding these calculations is essential for conservation efforts to sustainably manage natural resources like fish populations.